Category A:

• Henry VI, Part I
• Henry VI, Part II
• Taming of the Shrew (arguably)
• The Two Gentlemen of Verona
• Edward III
• The Merchant of Venice
• Henry IV, Part I
• Henry V
• The Merry Wives of Windsor
• Pericles
• Cymbeline
• Henry VIII

Category A½:

• The Two Noble Kinsmen

Category B:

• Henry VI, Part III
• Richard III
• Titus Andronicus
• Romeo and Juliet
• Richard II
• King John
• Henry IV, Part II
• Julius Caesar
• Othello
• King Lear
• Macbeth
• Antony and Cleopatra
• Coriolanus
• Timon of Athens

Category C:

• The Comedy of Errors
• Love’s Labour’s Lost
• A Midsummer Night’s Dream
• Much Ado About Nothing
• As You Like It
• Twelfth Night
• All’s Well That Ends Well
• Measure for Measure
• The Winter’s Tale
• The Tempest

Actually I’ll explain tomorrow in the unlikely event that nobody does so in comments. Feel free to say how obvious the answer is, and to point out the plays I have miscategorised above.

Edited to add: Yeah, Hamlet should have been listed under Category B!